A Tannakian Description for Parahoric Bruhat-Tits Group Schemes

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Wilson, Kevin Michael
Haines, Thomas
Let K be a field which is complete with respect to a discrete valuation and let O be the ring of integers in K. We study the Bruhat-Tits building B(G) and the parahoric Bruhat-Tits group schemes G<sub>F</sub> associated to a connected reductive split linear algebraic group G defined over O. In order to study these objects we use the theory of Tannakian duality, developed by Saavedra Rivano, which shows how to recover G from its category of finite rank projective representations over O. We also use Moy-Prasad filtrations in order to define lattice chains in any such representation. Using these two tools, we give a Tannakian description to B(G). We also define a functor Aut<sub>F</sub> associated to a facet F in B(G) in terms of lattice chains in a Tannakian way. We show that Aut<sub>F</sub> is representable by an affine group scheme of finite type, has the same generic fiber as G, and satisfies Aut<sub>F</sub>(O<sub>E</sub>) = G<sub>F</sub> (O<sub>E</sub>) for every unramified Galois extension E of K. Moreover, we show that there is a canonical morphism from G<sub>F</sub> to Aut<sub>F</sub>, which we conjecture to be an isomorphism. We prove that it is an isomorphism when the residue characteristic of K is zero and G is arbitrary, when G = GL<sub>n</sub> and K is arbitrary, and when F is the minimal facet containing the origin and G and K are arbitrary.